Question 81
Let $ABC$ be a triangle with altitudes $AD$, $BF$ and $CE$. Let $P$ and $Q$ be two points on line $EF$ such that $EP = DF$ ($E$ is between $P$ and $F$) and $QF = DE$ ($F$ is between $E$ and $Q$). Assuming that the perpendicular bisector of $DQ$ intersects $AB$ at $X$ and the perpendicular bisector of $DP$ intersects $AC$ at $Y$, prove that the midpoint of $BC$ lies on line $XY$.